Small-Signal Modeling and Comparisons of the Pulse Frequency Modulation and Time Shift Control for the LLC Converters

This article describes a new method for the small-signal modeling of the LLC converter based on time-domain analysis. The time-domain expressions of the state trajectory are transformed to the first-order of Tylor's expressions, which can achieve the linearization of the trajectory expressions. The state space formulas and transfer functions in pulse frequency modulation (PFM) are derived from the linearization expressions, where the state variables include the initial resonant current and resonant capacitor voltage in a switching period besides the output voltage. This method can be easily extended to the small signal modeling of the LLC converter in time-shift control (TSC). The reason why the TSC is regarded as a current-mode control is analytically explained in this article. The comparison of the frequency response in the PFM and TSC demonstrates the high bandwidth and high phase margin in the TSC, which indicates the LLC converter with the TSC can achieve a good dynamic response. The closed-loop PI controller is designed based on the small-signal model, and load step experiments demonstrate that the converter exhibits better dynamic response in the TSC compared to the PFM. The detailed derivation of the small signal modeling and the Matlab code for the modeling can be accessed at https://github.com/Guozq32/LLC-small-signal-modeling.